Signals & Systems Test - 2 - PDF Flipbook
Signals & Systems Test - 2
311 Views
2 Downloads
PDF 4,484,601 Bytes
GATE
EEE
Signals
&
Systems
Test-02Solutions
SIGNALS & SYSTEMS
1. Which one of the following input-output relationships is that of
a linear system?
Answer: (d)
Solution:
A linear system, in continuous time or discrete time, is a system
that possesses the important property of superposition.
A direct consequence of the superposition property is that, for
linear systems, an input which is zero for all time results in an
output which is zero for all time.
Note: Let y1(t) and Y2(t) be the responses of a continuous time
system to the inputs x1(t) and x2(t) respectively. Then the system
is linear if:
1
(i) The response to x1(t) +x2(t) is y1(t)+y2(t).
⟹ Additivity property.
(ii) The response to ax1(t) is ay1(t), where a is any complex
constant.
⟹ Scaling or homogeneity property.
1; = 1
2. A signal V(n) is defined by V[n] = � −1; = −1 which
0; = 0 | | > 1
is the value of composite signal defined as V[n] + V [ – n]?
a) 0 for all integer values of n
b) 2 for all integer values of n
c) 1 for all integer values of n
d) – 1 for all integer values of n
Answer: (a)
Solution:
V[n] +V [– n] = 0
For all integer values of n
2
3. The unit sample response of a discrete system is1 ½ ¼ 0 0 0....
For an input sequence 1 0 1 0 0 0 ...., what is the output
sequence?
a) 1 ½ ¼ ½ ¼ 0 0 …
b) 1 0 ¼ 0 0 …
c) 2 ½ 5/4 0 0 …
d) 1 ½ 5/4 ½ ¼ 0 0 …
Answer: (d)
Solution:
Y(n) = x(n)*h(n)
By add and shift method
Output sequence is
Y(n) = {1 1 �1 + 14� 1 1 0 0….
2 2 4
= {1 1 5 1 1 0 0….
2 4 2 4
4. A discrete – time signal x[n] = sin( 2n), n being an integer is
a) periodic with period π
b) periodic with period π2
c) periodic with period is π/2
3
d) not periodic.
Answer: (d)
Solution:
For discrete signal to be periodic 0 ratio must be a rational
2
number.
0 = 2
0 = ⁄2 is a irrational number, so x(n) is non periodic
2
5. The autocorrelation function of an energy signal has
a) no symmetry
b) conjugate symmetry
c) odd symmetry
d) even symmetry
Answer: (b)
Solution:
Auto correlation function of an energy signal g(t) has conjugate
symmetry ( ) = *(− )
6. For half-wave (odd) symmetry, with T0 = period of x(t), which
one of the following is correct?
a) X (t ± 0⁄2) = – x(t)
b) X (t ± 0⁄2) = x(t)
c) X (t ± 0) = – x(t)
d) X (t ± 0) = x(t)
Answer: (a)
4
Solution:
If x� ± 2 0� = – x(t) then x(t) have half-wave odd symmetry.
7. Match List-I (CT Function) with List-II (CT Fourier
Transform) and select the correct answer using the code given
below the lists:
List-I List-II
A. − ( ) 1. 2
1+ 2
B. ( ) = �10,, | | ≤ 1 2. j ( )
| | > 1
C. ( ) 3. 1
1+
D. 2 4. 2 sin
1+ 2
Codes:
A BCD
a) 1 4 2 3
b) 3 2 4 1
a) 1 2 4 3
b) 3 4 2 1
Answer: (d)
8. Let h (t) is the impulse response of a linear time invariant
system. Then the response of the system for any input u(t) is
a) ∫0 ℎ( ) ( − )
b) ∫0 ℎ( ) ( − )
c) ∫0 �∫0 ℎ( ) ( − ) �dt
5
d) ∫0 ℎ2( ) ( − )
Answer: (a)
Solution:
Consider the system shown in fig.1
Note that here u(t) is not a unit step function. It is any arbitrary
input.
For LTI system y(t) = u(t)*h(t)
Y(t) =∫ ∞ =−∞ ℎ( ) ( − ) or ∫ ∞ =−∞ ℎ( − ) ( )
=∫ ∞ =0 ℎ( ) ( − ) ,
if the system is causal i.e., h(t) = 0 for t < 0.
∫ = 0 ℎ( ) ( − ) ,
if the system is causal and also the input u(t) is causal
i.e., U(t) = 0 for t < 0.
9. Consider the following transforms:
1. Fourier transform
2. Laplace transform
Which of the above transforms is/are used in signal processing?
a) 1 only
b) 2 only
c) Both 1 and 2
d) Neither 1 nor 2
6
Answer: (a)
Solution:
• Laplace transform used for stability verifications, transient
analysis and system synthesis.
• In signal processing (which basically means filtering) fourier
transform are used, as filtering requires information purely in
terms of frequency.
10. What is the Laplace transforms of the waveform shown below?
a) F(S) = 1 + 1 − − 2 −2
b) F(S) = 1 − 1 − + 2 −2
c) F(S) = 1 + 1 + 2 2
d) F(S) = 1 − 1 − − 2 −
Answer: (a)
Solution:
f(t) = u(t) + u(t – 1) – 2u(t – 2)
F(s) = 1 + − − 2 −2
7
11. If F(s) and G(s) are the Laplace transform of f(t) and g(t), then
their product F(s) G(s) = H(s), where H(s) is the Laplace
transform of h(t), is defined as
a) (f∙g) (t)
b) ∫0 ( ) ( − )
c) Both A and B are correct
d) F(t)∙ g(t)
Answer: (b)
Solution:
This is a convolution property in the time domain
y(t) = ∫0 ( )g(t – ) d
Note: As per RTI filed and reply of UPSC answer is option (c).
12. A band limited signal is sampled at the Nyquist rate. The
signal can be recovered by passing the samples through
a) an RC filters
b) an envelope detector
c) a PLL
d) an ideal low-pass filter with the appropriate bandwidth
Answer: (d)
Solution:
The spectrum of a signal x(t) band limited to w is shown in
fig.1.
8
If x(t) is sampled spectrum (f) is shown in FIG.2.
( ) = ∑ ∞ =−∞ ( − )
The spectrum X(f) and hence x(t) can be recovered by passing
the samples through an ideal LPF of transfer function H(f)
shown in fig.3 with BW =W.
13. The input-output relationship of a causal stable LTI system is
given as y[n] = ∝y [n – t] + x[n]. If the impulse response h[n]
of this system satisfies the condition in ∑∞ =0 ℎ[ ] = 2, the
Relationship between α and is
a) α = 1−
2
b) α = 1+
2
c) α = 2β
9
d) α = – 2β
Answer: (a)
Solution:
H(z) =
1− −1
( )| =1 = 2
= 2 ⟹ = 1 −
1− 2
= 1 −
2
14. Assertion (A): There are no convergence issues with the
discrete-time Fourier series in general.
Reason (R): A discrete-time signal is always obtained by
sampling a continuous-time signal.
a) Both A and R are true and R is the correct explanation of A
b) Both A and R are true but R is NOT the correct explanation
of A
c) A is true but R is false
d) A is false but R is true
Answer: (d)
Solution:
Convergence is an important issue with the discrete –time
Fourier series in general as discrete – time Fourier series exists
only for convergent signals.
10
15. The impulse response of a discrete system with a simple pole
is shown in the given figure
The pole must be located
a) on the real axis at z = 1
b) on the real axis at z = – 1
c) at the origin of the z-plane
d) at z = ∞
Answer: (b)
Solution:
From diagram we have impulse response,
H(n) = (−1) u(n)
= 1
[1—1) −1
= 1
1+ −1
Z=–1
16. The z - transform of the time function ∑ ∞ =0 ( − )is
a) (Z – 1)/Z
b) Z/( − 1)2
c) Z / (Z – 1)
d) ( − 1)2/Z
Answer: (c)
11
Solution:
Let the given time function be
X(n) = ∑∞ =0 ( − )
X(n) = δ(n) + δ (n – 1) + δ (n – 2) +…= u(n), where u(n) is the
unit sample sequence or unit step sequence
⸫X(z) = Z.T of [u(n)] = − 1, | | > 1
Remember that region of convergence (ROC) is to be specified
for any X(z).
17. Which one of the following is the correct statement?
The region of convergence of z-transform of x[n] consists of the
values of z for which x[n]r –n is
a) absolutely integrable
b) absolutely summable
c) unity
d) < 1
Answer: (b)
18. Assertion (A): The system function H(Z) = 3 − 2 2 + is not
41 1
2 + + 8
causal.
Reason (R): If the numerator of H(z) is of lower order than the
denominator, the system may be causal.
a) Both A and R are true and R is the correct explanation of A
b) Both A and R are true but R is NOT the correct explanation
of A
c) A is true but R is false
12
d) A is false but R is true
Answer: (a)
Solution:
For causality of H(z), degree of numerator must be less than the
degree of denominator along with other conditions.
19. An ideal low-pass filter has a cut-off frequency of 100 Hz. If
the input to the filter in volts is V(t) = 30√2 sin 1256 , the
magnitude of the output of the filter will be
a) 0V
b) 100 V
c) 20 V
d) 200 V
Answer: (a)
Solution:
An ideal low pass filter has a cut –off frequency of 100 Hz, so it
passes the signals which have their frequency components
below or equal to 100Hz. When the signal of 200 Hz (as given
in the problem) will try to pass through LPF, it will not pass and
hence the output will be zero.
20. The Fourier Transform of the signal x(t) = −3 2 is of the
following form, where A and B are constants
a) A − | |
b) A −
c) A + B| |2
d) A − 2
13
Answer: (d)
Solution:
The Fourier transform of a normalized Gaussian pulse is also a
normalized Gaussian Pulse.
g(t) = − 2 → G(f) = − 2
⸫X(t) = −3 2 → G(f) = − 2
The constants A &B can be found if necessary by using time
scaling property:
If g(t) →G(f)
G(at) → 1 G� �
| |
For g(t) = − 2
G(at) = − 2 2
If X(t) = −3 2 = g(at) = − 2 2
−3 2 = − 2 2,
π 2 = 3, a =� 3 ,
−3 2 → � 3 − 3 2 = � 3 − 3 2 2
⸫A =� 3 , B = 2
3
General Formula:
K − 2 → KA − 2
Where A =� , B = 2
14
21. Input x(t) and output y(t) of an LTI system are related by the
differential equation "( ) − ′( ) −6y(t) = x(t). If the system
is neither causal nor stable, the impulse response h(t) of the
system is
a) 1 3 (− ) + 1 −2 (− )
5 5
b) −1 3 (− ) + 1 −2 (− )
5 5
c) 1 3 (− ) + 1 −2 ( )
5 5
d) − 1 3 (− ) − 1 −2 ( )
5 5
Answer: (b)
Solution:
2 ( ) − ( ) − 6y(t) = x(t)
2
2y(s) – s Y(s) – 6y(s) = x(s)
H(s) = 1 = 1
2− −6 ( −3)( +2)
H(s) = 1⁄5 − 1⁄5
−3 +2
System neither causal nor stable
So ROC is < −2 = ( < 3) ∩ ( < −2)
H(t) = − 1 3 (− ) + 1 −2 (− )
5 5
22. The system with characteristic equation s4+3s3+6s2+12 = 0
a) Stable
b) Unstable
c) Marginally stable
d) Marginally unstable
15
Answer: (b)
Solution:
According to R-H array
Two sign change. So it has two poles in RHS.
So, system is unstable.
23. A point moving in the complex plane satisfies the following
relation Z2+ ∗2 = 8, where Z* stands for the complex conjugate
of z. The difference of the distance of the moving point from
(2√2, 0) and (-2√2, 0) is
a) 8
b) 2
c) 4
d) 6
Answer: (c)
24. If X(Z) = + −−13, then x(n) series has
+
a) alternate 0s
b) alternate 1s
c) alternate 2s
d) alternate –1s
Answer: (a)
16
Solution:
X(z) = + −3 = 1+ −4
+ −1 1+ −2
= (1 + −4)(1 + −2)−1
= (1 + −4)(1 − −2 + −4 − −6 + ⋯ )
=1− −2 + 2 −4 − 2 −6 +….
So, x(n) = (n) – δ(n – 2) + 2δ(n – 4) + 2 δ(n – 6) + ….
Hence, x(n) series has alternate zeros, i.e., at n = 1,3,5, …
25. Consider the equation Re(1/z) = C, where z is a complex
number, C is a nonzero constant and Re () represents the real
part. The equation describes a
a) Straight line
b) Parabola
c) Circle
d) None of these
Answer: (c)
Solution:
1 = 1 −
+ −
= −
2+ 2
Re� 1 � = = C
2+ 2
2 + 2 =
⟹ 2 + 2 − = 0
This is equation of pair of a circle.
17
26. If α and β are the roots of the equation x2 – px + q = 0, then
∑ α2
a) p2 + 2q
b) p + 2q
c) p2 – 2q
d) p – 2q
Answer: (c)
Solution:
x2 – px + q = 0
2 + 2 = ( + )2 −2αβ
= 2 − 2q
27. Number of real values of (a + ib)1/n + (a – ib)1/n is
a) 0
b) 1
c) N
d) None of these
Answer: (b)
28. Consider the following system function of a discrete-time LTI
system: H(Z) = −1− ∗ where, a* is the complex conjugate of a.
1− −1
The frequency response of such a system is
a) aperiodic; depends on frequency
b) aperiodic; does not depends on frequency
c) periodic; depends on frequency
d) periodic; does not depends on frequency
Answer: (b)
18
Solution:
Given, H(z) = −1 − ∗ Put z = r Ω
1 − −1
The discrete time frequency response will be aperiodic and does
not depends upon the frequency.
29. The Nyquist sampling interval, for the signal Sinc(700t) + Sinc
(500t) is
a) 1
350
b)
350
c) 1
700
d)
175
Answer: (c)
Let the given signal be
X(t) = 1( ) + 2( ) ….. (1)
1( ) = sin (700 ) → 1( )
2( ) = sin (500 ) → 2( )
1( ) and 2( ) are shown in Figures.
19
From (1), X(f) = 1( ) + 2( )
⸫ X(f) has highest frequency, ℎ = 350 Hz
(min) = Nyquist sampling frequency
= 700 HZ.
⸫ Nyquist sampling interval = (max) = 1/700 sec.
30. Consider the pulse shape s(t) as shown. The impulse response
h(t) of the filter matched to this pulse is
20
Answer: (c)
Solution:
Impulse response h(t) of the filter matched to the input pulse s(t)
exiting from t = 0 to t = T is given by h(t) = s[– (t – T)]
The operations of time reversal and then time delay of T should
be performed on s(t), to get h(t). This is equivalent to the
following statement:
Scan the sketch of s(t) backward from t = T to t = 0 and draw the
sketch of h(t) forward from t = 0 to t = T. s(t) & h(t) are shown
in Figure.
21
Click to View FlipBook Version Previous Book Signals & Systems Test - 1 Next Book Logical Reasoning Test - 4
Data Loading...